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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $G$.\nThen $n_p = [G : N_G(P)]$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$.\n\n## Remarks\n\nBy the [[second Sylow theorem|09GCQ5STVJ]], $G$ acts transitively by conjugation on $\\operatorname{Syl}_p(G)$.\nThe stabiliser of $P$ is $N_G(P)$, so the [[orbit-stabiliser theorem|0E2HSCKZ6C]] gives the formula.\nIn particular, $P$ is normal in $G$ if and only if $n_p = 1$.\n\n## Explanations\n\n### L1 human\n\nAll Sylow $p$-subgroups are conjugates of one of them, $P$.\nThe number of conjugates of $P$ is the number of cosets of the elements that do not move $P$.\nThese elements form the normaliser.\n",
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